**2. Essential notions and theoretical background**

In the introduction, we gave a substantive description of a fuzzy set; now we give its mathematical description.

**Definition 1.** An ordered pair f g *x*, *μ*ð Þ *x* , where*x*∈ *X*, *μ* : *X* ! ½ � 0, 1 is called a fuzzy set.

Here *X* is the universal set of real numbers (universe), *μ*ð Þ *x* is the membership function of fuzzy set, and *μ* : *X* ! ½ � 0, 1 means that the membership function takes values from the interval [0; 1] for all *x*.

An important special case of fuzzy sets is fuzzy numbers. A *fuzzy number* is a fuzzy subset of the universal set of real numbers that has a normal and convex membership function, that is, such that: (a) there is an element of the universe in which the membership function is equal to one, and also (b) when deviating from its maximum left or right, membership function does not increase.

In this chapter we will deal with trapezoidal fuzzy numbers. Almost all the results given in this and third sections are, with minor modifications, taken from [7]; therefore we will refrain from further citation.

We denote trapezoidal fuzzy numbers in *<sup>X</sup>* by *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> ð Þ *<sup>a</sup>*, *<sup>b</sup>*,*c*, *<sup>d</sup>* , 0<sup>&</sup>lt; *<sup>a</sup>*<sup>≤</sup> *<sup>b</sup>*<sup>≤</sup> *<sup>c</sup>*≤*d*. The membership function's graph is a trapezoid with vertices (*a*; 0), (*b*; 1), (*c*; 1), and (*d*; 0).

We denote by <sup>Ψ</sup>ð Þ¼ *<sup>X</sup> <sup>R</sup>*~*<sup>i</sup>* <sup>¼</sup> *ai*, *bi*,*ci* ð Þ , *di* , *<sup>i</sup>* <sup>∈</sup> the set of all trapezoidal fuzzy numbers in the universe *X*.

The determinations of some operations on trapezoidal fuzzy numbers are given below.

$$
\bar{R}\_1 = \bar{R}\_2 \Leftrightarrow a\_1 = a\_2,\\
b\_1 = b\_2,\\
c\_1 = c\_2,\\
d\_1 = d\_2, \ \bar{R}\_1, \bar{R}\_2 \in \Psi(X). \tag{1}
$$

$$
\tilde{R}\_1 \oplus \tilde{R}\_2 = (a\_1 + a\_2, b\_1 + b\_2, c\_1 + c\_2, d\_1 + d\_2), \quad \tilde{R}\_1, \tilde{R}\_2 \in \Psi(X). \tag{2}
$$

$$a \otimes \tilde{R} = (aa, ab, ac, ad) \text{ } a > 0, \text{ } \tilde{R} \in \Psi(X). \tag{3}$$

**Definition 2.** Trapezoidal fuzzy number *<sup>R</sup>*~<sup>1</sup> <sup>¼</sup> ð Þ *ai* is included in trapezoidal fuzzy number *<sup>R</sup>*~<sup>2</sup> <sup>¼</sup> ð Þ *bi* , *<sup>i</sup>* <sup>¼</sup> 1, 4, i*:*e*:R*~<sup>1</sup> <sup>⪯</sup> *<sup>R</sup>*~2, if and only if

$$a\_1 \le a\_2, b\_1 \le b\_2, c\_1 \le c\_2, d\_1 \le d\_2 \tag{4}$$

Suppose we have the finite collection of trapezoidal fuzzy numbers:

*A New Approach for Assessing Credit Risks under Uncertainty*

*DOI: http://dx.doi.org/10.5772/intechopen.93285*

*R*~<sup>1</sup> 7 7.5 8 8.8 *R*~<sup>2</sup> 6 6.1 7.7 9 *R*~<sup>3</sup> 1 3 5 9.5 *R*~<sup>4</sup> 7.6 7.9 8.1 8.9

Compare with it the following finite collection of trapezoidal fuzzy numbers:

<sup>0</sup> 1 3 5 8.8

<sup>0</sup> 6 6.1 7.7 8.9

<sup>0</sup> 7 7.5 8 9

<sup>0</sup> 7.6 7.9 8.1 9.5

We see that the matching columns in both tables consist of equal sets; at the

0

' , *c*<sup>1</sup> <sup>0</sup> ≤ *c*<sup>2</sup>

*j*¼1

tatives of finite collection of trapezoidal fuzzy numbers and its regulation coincide. It is obvious that the regulation represents a finite collection of nested trapezoi-

The following theorem yields a formal definition of a representative.

*<sup>m</sup>*, *m* ¼ 2, 3, … *:*.

**Theorem 1** [7]**.** *In the metric space of trapezoidal fuzzy numbers, the representative*

*ρ* ~ *S*, *R*~<sup>0</sup> *j*

� �, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … . From Eq. (11) it follows that represen-

}, and {*dj*} and {*dj*

0

*S*∈ Ψð Þ *X* and the finite collection of trapezoidal

*<sup>m</sup>=*2þ<sup>1</sup> *if m is even*; (12)

ð Þ *<sup>m</sup>*þ<sup>1</sup> *<sup>=</sup>*<sup>2</sup> *if m is odd:* (13)

, *d*<sup>1</sup> <sup>0</sup> ≤ *d*<sup>2</sup>

� � (11)

� �, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, …, *is*

<sup>0</sup> ≤ … ≤ *cm*<sup>0</sup>

same time the elements of the sets in the second table form nondecreasing sequences. By the regulation of the finite collection of trapezoidal fuzzy numbers

n o. The strict definition of regulation will be given below.

regulation of the finite collection of trapezoidal fuzzy numbers *R*~ *<sup>j</sup>*

0

X*m j*¼1

<sup>2</sup> <sup>⪯</sup> … <sup>⪯</sup> *<sup>R</sup>*~<sup>0</sup>

*<sup>m</sup>=*<sup>2</sup> <sup>⪯</sup> *<sup>R</sup>*<sup>~</sup> <sup>∗</sup> <sup>⪯</sup> *<sup>R</sup>*~<sup>0</sup>

*<sup>R</sup>*<sup>~</sup> <sup>∗</sup> <sup>¼</sup> *<sup>R</sup>*~<sup>0</sup>

*<sup>R</sup>*<sup>~</sup> <sup>∗</sup> *of the finite collection of trapezoidal fuzzy numbers*, *<sup>R</sup>*<sup>~</sup> *<sup>j</sup>*

*R*~0

**Definition 4.** The finite collection of trapezoidal fuzzy numbers *R*~<sup>0</sup>

}, {*cj*} and {*cj*

Due to this definition and Eq. (9), it is obvious that the equality

*ρ* ~ *S*, *R*~ *<sup>j</sup>* � � <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*

<sup>0</sup> ≤ … ≤ *bm*

� �, we will mean the finite collection of trapezoidal fuzzy numbers

*a'*

*R*~1

*R*~2

*R*~3

*R*~4

*R*~1,*R*~2,*R*~3,*R*~<sup>4</sup>

sets {*aj*} and {*aj*

fuzzy numbers *R*~ *<sup>j</sup>*

dal fuzzy numbers: *R*~<sup>0</sup>

*determined as follows:*

**185**

0

and *j* ¼ 1, *m*, *m* ¼ 2, 3, … .

<sup>0</sup> ≤ … ≤ *am*<sup>0</sup>

}, {*bj*} and {*bj*

holds in the metric space for any ~

<sup>1</sup> <sup>⪯</sup> *<sup>R</sup>*~<sup>0</sup>

, *b*<sup>1</sup> ' ≤ *b*<sup>2</sup>

*R*~0 1,*R*~<sup>0</sup> 2,*R*~<sup>0</sup> 3,*R*~<sup>0</sup> 4

and *a*<sup>1</sup> ' ≤ *a*<sup>2</sup> *<sup>j</sup> b'*

*aj bj cj dj*

*<sup>j</sup> c*

*'*

*<sup>j</sup> d'*

*j* n o is a

� � if the finite

} are pairwise equal

<sup>0</sup> ≤ … ≤ *dm*<sup>0</sup>

,

*j*

It is known that fuzzy maximum and minimum of two trapezoidal fuzzy numbers is defined as follows [8]:

$$\begin{cases} \widehat{\text{max}}\left\{\tilde{R}\_1, \tilde{R}\_2\right\} = (\max\left\{a\_1, b\_1\right\}, \max\left\{a\_2, b\_2\right\}, \max\left\{a\_3, b\_3\right\}, \max\left\{a\_4, b\_4\right\}),\\ \widehat{\text{min}}\left\{\tilde{R}\_1, \tilde{R}\_2\right\} = (\min\left\{a\_1, b\_1\right\}, \min\left\{a\_2, b\_2\right\}, \min\left\{a\_3, b\_3\right\}, \min\left\{a\_4, b\_4\right\}). \end{cases} \tag{5}$$

Hence it follows that the above definition is equivalent to those given in the literature (see, e.g., [9, 10]):

$$
\bar{R}\_1 \le \bar{R}\_2 \Leftrightarrow \begin{cases}
\widehat{\min}\left\{\bar{R}\_1, \bar{R}\_2\right\} = \bar{R}\_1, & \bar{R}\_1, \bar{R}\_2 \in \Psi(X). \\
\widehat{\max}\left\{\bar{R}\_1, \bar{R}\_2\right\} = \bar{R}\_2,
\end{cases} \tag{6}
$$

Now we are going to introduce a metric on Ψ(*X*) , i.e., define a distance between trapezoidal fuzzy numbers.

We say that the function *v:* <sup>Ψ</sup> (*X*)! <sup>ℜ</sup><sup>+</sup> is *isotone valuation on* <sup>Ψ</sup>(*X*) if

$$\nu\left(\widehat{\mathbf{m}}\overline{\mathbf{x}}\left\{\tilde{R}\_1,\tilde{R}\_2\right\}\right) + \nu\left(\widehat{\mathbf{m}}\overline{\mathbf{n}}\left\{\tilde{R}\_1,\tilde{R}\_2\right\}\right) = \nu\left(\tilde{R}\_1\right) + \nu\left(\tilde{R}\_2\right) \tag{7}$$

and

$$
\tilde{R}\_1 \preceq \tilde{R}\_2 \Rightarrow \upsilon\left(\tilde{R}\_1\right) \leq \upsilon\left(\tilde{R}\_2\right). \tag{8}
$$

The isotone valuation *v* determines *the metric* on Ψ (*X*):

$$\rho\left(\tilde{R}\_1,\tilde{R}\_2\right) = v\left(\text{m}\overline{\text{ax}}\left\{\tilde{R}\_1,\tilde{R}\_2\right\}\right) - v\left(\text{m}\overline{\text{hin}}\left\{\tilde{R}\_1,\tilde{R}\_2\right\}\right) \tag{9}$$

Ψ(*X*) with isotone valuation *v* and metric (Eq. (8)) is called a *metric space* of trapezoidal fuzzy numbers.

**Definition 3.** In the metric space, the trapezoidal fuzzy number *R*~ <sup>∗</sup> is*the representative* of the finite collection of trapezoidal fuzzy numbers *R*~ *<sup>j</sup>* � �, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … if

$$\sum\_{j=1}^{m} \rho\left(\tilde{\mathcal{R}}^{\*}, \tilde{\mathcal{R}}\_{j}\right) \le \sum\_{j=1}^{m} \rho\left(\tilde{\mathcal{S}}, \tilde{\mathcal{R}}\_{j}\right), \ \forall \tilde{\mathcal{S}} \in \Psi(\mathcal{X}) \tag{10}$$

Let us clarify the meaning of this definition. A representative of the given finite collection of trapezoidal fuzzy numbers is a trapezoidal fuzzy number such that the sum of the distances between it and all members of this collection is minimal.

For an accommodation of posterior theoretical constructions, we need to introduce a concept of *regulation* of finite collection of trapezoidal fuzzy numbers. We begin with an example.

*A New Approach for Assessing Credit Risks under Uncertainty DOI: http://dx.doi.org/10.5772/intechopen.93285*

*<sup>R</sup>*~<sup>1</sup> <sup>¼</sup> *<sup>R</sup>*~<sup>2</sup> <sup>⇔</sup> *<sup>a</sup>*<sup>1</sup> <sup>¼</sup> *<sup>a</sup>*2, *<sup>b</sup>*<sup>1</sup> <sup>¼</sup> *<sup>b</sup>*2,*c*<sup>1</sup> <sup>¼</sup> *<sup>c</sup>*2, *<sup>d</sup>*<sup>1</sup> <sup>¼</sup> *<sup>d</sup>*2, *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup> <sup>∈</sup> <sup>Ψ</sup>ð Þ *<sup>X</sup> :* (1) *<sup>R</sup>*~<sup>1</sup> <sup>⊕</sup> *<sup>R</sup>*~<sup>2</sup> <sup>¼</sup> ð Þ *<sup>a</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*2, *<sup>b</sup>*<sup>1</sup> <sup>þ</sup> *<sup>b</sup>*2,*c*<sup>1</sup> <sup>þ</sup> *<sup>c</sup>*2, *<sup>d</sup>*<sup>1</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup> , *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup> <sup>∈</sup> <sup>Ψ</sup>ð Þ *<sup>X</sup> :* (2)

**Definition 2.** Trapezoidal fuzzy number *<sup>R</sup>*~<sup>1</sup> <sup>¼</sup> ð Þ *ai* is included in trapezoidal

It is known that fuzzy maximum and minimum of two trapezoidal fuzzy

� � <sup>¼</sup> ð Þ max f g *<sup>a</sup>*1, *<sup>b</sup>*<sup>1</sup> , max f g *<sup>a</sup>*2, *<sup>b</sup>*<sup>2</sup> , max f g *<sup>a</sup>*3, *<sup>b</sup>*<sup>3</sup> , max f g *<sup>a</sup>*4, *<sup>b</sup>*<sup>4</sup> ,

� � <sup>¼</sup> ð Þ min f g *<sup>a</sup>*1, *<sup>b</sup>*<sup>1</sup> , min f g *<sup>a</sup>*2, *<sup>b</sup>*<sup>2</sup> , min f g *<sup>a</sup>*3, *<sup>b</sup>*<sup>3</sup> , min f g *<sup>a</sup>*4, *<sup>b</sup>*<sup>4</sup> *:*

Hence it follows that the above definition is equivalent to those given in the

� � <sup>¼</sup> *<sup>R</sup>*~1,

� � <sup>¼</sup> *<sup>R</sup>*~2,

Now we are going to introduce a metric on Ψ(*X*) , i.e., define a distance between

<sup>g</sup> *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup> � � � � <sup>¼</sup> *<sup>v</sup> <sup>R</sup>*~<sup>1</sup>

� � � � � *<sup>v</sup>* min

**Definition 3.** In the metric space, the trapezoidal fuzzy number *R*~ <sup>∗</sup> is*the representa-*

*ρ* ~ *S*, *R*~ *<sup>j</sup>* � �, ∀~

Let us clarify the meaning of this definition. A representative of the given finite collection of trapezoidal fuzzy numbers is a trapezoidal fuzzy number such that the sum of the distances between it and all members of this collection is minimal.

For an accommodation of posterior theoretical constructions, we need to introduce a concept of *regulation* of finite collection of trapezoidal fuzzy numbers. We

Ψ(*X*) with isotone valuation *v* and metric (Eq. (8)) is called a *metric space* of

<sup>≤</sup> <sup>X</sup>*<sup>m</sup> j*¼1

� �≤*v R*~<sup>2</sup>

fuzzy number *<sup>R</sup>*~<sup>2</sup> <sup>¼</sup> ð Þ *bi* , *<sup>i</sup>* <sup>¼</sup> 1, 4, i*:*e*:R*~<sup>1</sup> <sup>⪯</sup> *<sup>R</sup>*~2, if and only if

numbers is defined as follows [8]:

max <sup>g</sup> *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup>

literature (see, e.g., [9, 10]):

trapezoidal fuzzy numbers.

trapezoidal fuzzy numbers.

begin with an example.

**184**

*R*~<sup>1</sup> ⪯ *R*~<sup>2</sup> ⇔

*<sup>v</sup>* max <sup>g</sup> *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup>

*ρ R*~1, *R*~<sup>2</sup>

X*m j*¼1 min <sup>g</sup> *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup>

(

� � � � <sup>þ</sup> *<sup>v</sup>* min

The isotone valuation *v* determines *the metric* on Ψ (*X*):

� � <sup>¼</sup> *<sup>v</sup>* max <sup>g</sup> *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup>

*tive* of the finite collection of trapezoidal fuzzy numbers *R*~ *<sup>j</sup>*

*ρ R*~ <sup>∗</sup> , *R*~ *<sup>j</sup>* � �

max <sup>g</sup> *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup>

We say that the function *v:* <sup>Ψ</sup> (*X*)! <sup>ℜ</sup><sup>+</sup> is *isotone valuation on* <sup>Ψ</sup>(*X*) if

*<sup>R</sup>*~<sup>1</sup> <sup>⪯</sup> *<sup>R</sup>*~<sup>2</sup> ) *<sup>v</sup> <sup>R</sup>*~<sup>1</sup>

min <sup>g</sup> *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup>

*Banking and Finance*

(

and

*<sup>α</sup>* <sup>⊗</sup> *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> ð Þ *<sup>α</sup>a*, *<sup>α</sup>b*, *<sup>α</sup>c*, *<sup>α</sup><sup>d</sup> <sup>α</sup>* <sup>&</sup>gt;0, *<sup>R</sup>*<sup>~</sup> <sup>∈</sup> <sup>Ψ</sup>ð Þ *<sup>X</sup> :* (3)

*a*<sup>1</sup> ≤ *a*2, *b*<sup>1</sup> ≤ *b*2,*c*<sup>1</sup> ≤*c*2, *d*<sup>1</sup> ≤*d*<sup>2</sup> (4)

(5)

(9)

*<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup> <sup>∈</sup> <sup>Ψ</sup>ð Þ *<sup>X</sup> :* (6)

� � (7)

� � <sup>þ</sup> *<sup>v</sup> <sup>R</sup>*~<sup>2</sup>

<sup>g</sup> *<sup>R</sup>*~1, *<sup>R</sup>*~<sup>2</sup> � � � �

� �*:* (8)

� �, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … if

*S*∈ Ψð Þ *X* (10)


Suppose we have the finite collection of trapezoidal fuzzy numbers:

Compare with it the following finite collection of trapezoidal fuzzy numbers:


We see that the matching columns in both tables consist of equal sets; at the same time the elements of the sets in the second table form nondecreasing sequences. By the regulation of the finite collection of trapezoidal fuzzy numbers *R*~1,*R*~2,*R*~3,*R*~<sup>4</sup> � �, we will mean the finite collection of trapezoidal fuzzy numbers *R*~0 1,*R*~<sup>0</sup> 2,*R*~<sup>0</sup> 3,*R*~<sup>0</sup> 4 n o. The strict definition of regulation will be given below.

**Definition 4.** The finite collection of trapezoidal fuzzy numbers *R*~<sup>0</sup> *j* n o is a regulation of the finite collection of trapezoidal fuzzy numbers *R*~ *<sup>j</sup>* � � if the finite sets {*aj*} and {*aj* 0 }, {*bj*} and {*bj* 0 }, {*cj*} and {*cj* 0 }, and {*dj*} and {*dj* 0 } are pairwise equal and *a*<sup>1</sup> ' ≤ *a*<sup>2</sup> <sup>0</sup> ≤ … ≤ *am*<sup>0</sup> , *b*<sup>1</sup> ' ≤ *b*<sup>2</sup> <sup>0</sup> ≤ … ≤ *bm* ' , *c*<sup>1</sup> <sup>0</sup> ≤ *c*<sup>2</sup> <sup>0</sup> ≤ … ≤ *cm*<sup>0</sup> , *d*<sup>1</sup> <sup>0</sup> ≤ *d*<sup>2</sup> <sup>0</sup> ≤ … ≤ *dm*<sup>0</sup> , and *j* ¼ 1, *m*, *m* ¼ 2, 3, … .

Due to this definition and Eq. (9), it is obvious that the equality

$$\sum\_{j=1}^{m} \rho\left(\tilde{\mathbf{S}}, \tilde{\mathbf{R}}\_{j}\right) = \sum\_{j=1}^{m} \rho\left(\tilde{\mathbf{S}}, \tilde{\mathbf{R}}'\_{j}\right) \tag{11}$$

holds in the metric space for any ~ *S*∈ Ψð Þ *X* and the finite collection of trapezoidal fuzzy numbers *R*~ *<sup>j</sup>* � �, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … . From Eq. (11) it follows that representatives of finite collection of trapezoidal fuzzy numbers and its regulation coincide.

It is obvious that the regulation represents a finite collection of nested trapezoidal fuzzy numbers: *R*~<sup>0</sup> <sup>1</sup> <sup>⪯</sup> *<sup>R</sup>*~<sup>0</sup> <sup>2</sup> <sup>⪯</sup> … <sup>⪯</sup> *<sup>R</sup>*~<sup>0</sup> *<sup>m</sup>*, *m* ¼ 2, 3, … *:*.

The following theorem yields a formal definition of a representative.

**Theorem 1** [7]**.** *In the metric space of trapezoidal fuzzy numbers, the representative <sup>R</sup>*<sup>~</sup> <sup>∗</sup> *of the finite collection of trapezoidal fuzzy numbers*, *<sup>R</sup>*<sup>~</sup> *<sup>j</sup>* � �, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, …, *is determined as follows:*

$$
\bar{\mathcal{R}}'\_{m/2} \le \bar{\mathcal{R}}^\* \le \bar{\mathcal{R}}'\_{m/2+1} \text{ if } m \text{ is even;} \tag{12}
$$

$$
\tilde{\mathcal{R}}^{\*} = \tilde{\mathcal{R}}\_{(m+1)/2}' \text{ if } m \text{ is odd.}\tag{13}
$$

It follows from the theorem that when the number of members in a finite collection of trapezoidal fuzzy numbers is even, a representative can take on an infinite number of values. Now we introduce the specific aggregation operator that uniquely identifies the representative (here and further on, expression [*r*], where *r* is a real number, denotes the integer part of this number):

$$\bar{\boldsymbol{R}}^{\*} = \begin{cases} \left(\boldsymbol{a}\_{[m/2],i}^{\prime} + \boldsymbol{a}\_{[(m+3)/2],i}^{\prime}\right) / 2 \text{ if } \quad \sum\_{j=1}^{\lfloor (m+1)/2 \rfloor} \rho\left(\bar{\boldsymbol{\mathcal{R}}}\_{j}^{\prime}, \bar{\boldsymbol{\mathcal{R}}}\_{[m/2]}^{\prime}\right) = \sum\_{j=\lfloor m/2 \rfloor+1}^{\infty} \rho\left(\bar{\boldsymbol{\mathcal{R}}}\_{j}^{\prime}, \bar{\boldsymbol{\mathcal{R}}}\_{[(m+3)/2]}^{\prime}\right), & i = \overline{1,4} \\\ \boldsymbol{a}\_{[m/2],i}^{\prime} + \left(\frac{\sum\_{j=1}^{\lfloor (m+3)/2 \rfloor} \rho\left(\bar{\boldsymbol{\mathcal{R}}}\_{j}^{\prime}, \bar{\boldsymbol{\mathcal{R}}}\_{[m/2]}^{\prime}\right)}{\sum\_{j=1}^{\lfloor (m+3)/2 \rfloor} \rho\left(\bar{\boldsymbol{\mathcal{R}}}\_{j}^{\prime}, \bar{\boldsymbol{\mathcal{R}}}\_{[m/2]}^{\prime}\right) + \sum\_{j=\lfloor m/2 \rfloor+1}^{\infty} \rho\left(\bar{\boldsymbol{\mathcal{R}}}\_{j}^{\prime}, \bar{\boldsymbol{\mathcal{R}}}\_{[(m+3)/2]}^{\prime}\right) \boldsymbol{a}\_{[m/2],i}^{\prime} - \boldsymbol{a}\_{[m/2],i}^{\prime}}\right) \boldsymbol{a}\_{\left(m + \frac{3}{2}\right)} \end{cases} \tag{14}$$

**Remark 1.** It can be easily shown that the representative determined by Eq. (14) is a trapezoidal fuzzy number.

**Summary.** In this section, we presented the definitions of fuzzy sets and trapezoidal fuzzy numbers. Operations on trapezoidal fuzzy numbers are considered, and definitions of concepts that are necessary for constructing the proposed approach are given.

#### **3. Credit risk assessment method**

First of all, it is necessary to parameterize the risk assessment process, i.e., identify those parameters that to one degree or another affect credit risks. Such parameters may be credit history of the borrower, revenue, provision, market share, etc. The number and characteristics of risk assessment parameters are determined by an experienced lender manager.

As already mentioned in the introduction, there is always uncertainty in forecasting the values of the parameters for assessing credit risks, and, unfortunately, this fact cannot be completely avoided. An effective way out of this situation is to attract experts whose estimates are based on experience and intuition.

A very important point is the determination of a form for the submission of expert assessments. Here we present expert estimates in the form of trapezoidal fuzzy numbers. Let us justify our choice.

The expert has the opportunity to outline the following intervals (**Figure 1**):


Thus, the membership function of a trapezoidal fuzzy number is common to the

Now we need to aggregate expert assessments for each parameter and obtain the

Suppose a group of experts estimates the rating of an alternative under some given criterion. Though the experts are professionals of the same level, their subjective estimates may be essentially different. The problem consists in processing these estimates so that a consensus could be found. In constructing any kind of aggregation method under group decision-making, the key task is to determine the

Let us consider the finite collection of trapezoidal fuzzy numbers formed by

experts' estimates. To our mind, the representative of this collection, i.e., a trapezoidal fuzzy number such that the sum of distances between it and all other members of the given finite collection is minimal, is of particular interest.

cases considered and can easily be reduced to particular cases.

well-justified weights of importance for each expert.

result of group decision-making.

*General form of triangular fuzzy number.*

*General form of trapezoidal fuzzy number.*

*A New Approach for Assessing Credit Risks under Uncertainty*

*DOI: http://dx.doi.org/10.5772/intechopen.93285*

**Figure 2.**

**Figure 1.**

**Figure 3.**

**187**

*Segment of straight line y = 1.*

If, according to the expert's opinion, the parameter takes the maximum value at a single point, then *b* = *c*, and the estimate will take the form of a triangular fuzzy number (**Figure 2**).

If the expert is confident that the parameter reaches its maximum value in the interval [*b*, *c*], and its eastern values are nonpositive, then *a* = *b*, *c* = *d*, and the estimate takes the form of a segment of straight line *y* =1(**Figure 3**).

If in an extreme case the expert considers that the maximum value of the parameter is reached at a single point, and at all other points the values are zero, then *a* = *b* = *c* = *d*, and the graph of the membership function degenerates into the point with coordinates (*a*; 1).

*A New Approach for Assessing Credit Risks under Uncertainty DOI: http://dx.doi.org/10.5772/intechopen.93285*

**Figure 1.** *General form of trapezoidal fuzzy number.*

It follows from the theorem that when the number of members in a finite collection of trapezoidal fuzzy numbers is even, a representative can take on an infinite number of values. Now we introduce the specific aggregation operator that uniquely identifies the representative (here and further on, expression [*r*], where *r*

> *<sup>j</sup>*¼½ �þ *<sup>m</sup>=*<sup>2</sup> <sup>1</sup>*<sup>ρ</sup> <sup>R</sup>*~<sup>0</sup> *j* , *R*~<sup>0</sup> ½ � ð Þ *m*þ3 *=*2 � � *<sup>a</sup>*<sup>0</sup>

� � <sup>0</sup>

**Remark 1.** It can be easily shown that the representative determined by Eq. (14)

**Summary.** In this section, we presented the definitions of fuzzy sets and trapezoidal fuzzy numbers. Operations on trapezoidal fuzzy numbers are considered, and definitions of concepts that are necessary for constructing the proposed

First of all, it is necessary to parameterize the risk assessment process, i.e., identify those parameters that to one degree or another affect credit risks. Such parameters may be credit history of the borrower, revenue, provision, market share, etc. The number and characteristics of risk assessment parameters are

As already mentioned in the introduction, there is always uncertainty in forecasting the values of the parameters for assessing credit risks, and, unfortunately, this fact cannot be completely avoided. An effective way out of this situation is to

A very important point is the determination of a form for the submission of expert assessments. Here we present expert estimates in the form of trapezoidal

The expert has the opportunity to outline the following intervals (**Figure 1**):

• [a, b]—Where the parameter takes positive values, increasing from 0 to 1

• [c, d]—Where the parameter takes positive values, decreasing from 1 to 0

If, according to the expert's opinion, the parameter takes the maximum value at a single point, then *b* = *c*, and the estimate will take the form of a triangular fuzzy

If the expert is confident that the parameter reaches its maximum value in the interval [*b*, *c*], and its eastern values are nonpositive, then *a* = *b*, *c* = *d*, and the

If in an extreme case the expert considers that the maximum value of the parameter is reached at a single point, and at all other points the values are zero, then *a* = *b* = *c* = *d*, and the graph of the membership function degenerates into the

• [*b*, *c*]—The confidence interval where the parameter takes values 1

estimate takes the form of a segment of straight line *y* =1(**Figure 3**).

attract experts whose estimates are based on experience and intuition.

<sup>¼</sup> <sup>P</sup>*<sup>m</sup> j*¼½ �þ *m=*2 1 *ρ R*~<sup>0</sup> *j* , *R*~<sup>0</sup> ½ � ð Þ *m*þ3 *=*2 � �

,

½ � *m=*2 ,*i*

1 A*otherwise* *i* ¼ 1, 4

(14)

½ � ð Þ *<sup>m</sup>*þ<sup>3</sup> *<sup>=</sup>*<sup>2</sup> ,*<sup>i</sup>* � *a*<sup>0</sup>

is a real number, denotes the integer part of this number):

½ � ð Þ *<sup>m</sup>*P<sup>þ</sup><sup>1</sup> *<sup>=</sup>*<sup>2</sup> *j*¼1

P½ � ð Þ *<sup>m</sup>*þ<sup>1</sup> *<sup>=</sup>*<sup>2</sup> *<sup>j</sup>*¼<sup>1</sup> *<sup>ρ</sup> <sup>R</sup>*~<sup>0</sup> *j* , *R*~<sup>0</sup> ½ � *m=*2 � �

*ρ R*~<sup>0</sup> *j* , *R*~<sup>0</sup> ½ � *m=*2 � �

<sup>þ</sup> <sup>P</sup>*<sup>m</sup>*

*<sup>R</sup>*<sup>~</sup> <sup>∗</sup> <sup>¼</sup>

*a*0 ½ � *<sup>m</sup>=*<sup>2</sup> ,*<sup>i</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>0</sup>

*Banking and Finance*

8 >>>>>><

> *a*0 ½ � *<sup>m</sup>=*<sup>2</sup> ,*<sup>i</sup>* þ

>>>>>>:

½ � ð Þ *m*þ3 *=*2 ,*i* � �

> P½ � ð Þ *<sup>m</sup>*þ<sup>1</sup> *<sup>=</sup>*<sup>2</sup> *<sup>j</sup>*¼<sup>1</sup> *<sup>ρ</sup> <sup>R</sup>*~<sup>0</sup> *j* , *R*~<sup>0</sup> ½ � *m=*2 � �

**3. Credit risk assessment method**

determined by an experienced lender manager.

fuzzy numbers. Let us justify our choice.

@

is a trapezoidal fuzzy number.

approach are given.

number (**Figure 2**).

**186**

point with coordinates (*a*; 1).

*=*2 *if*

**Figure 2.** *General form of triangular fuzzy number.*

**Figure 3.** *Segment of straight line y = 1.*

Thus, the membership function of a trapezoidal fuzzy number is common to the cases considered and can easily be reduced to particular cases.

Now we need to aggregate expert assessments for each parameter and obtain the result of group decision-making.

Suppose a group of experts estimates the rating of an alternative under some given criterion. Though the experts are professionals of the same level, their subjective estimates may be essentially different. The problem consists in processing these estimates so that a consensus could be found. In constructing any kind of aggregation method under group decision-making, the key task is to determine the well-justified weights of importance for each expert.

Let us consider the finite collection of trapezoidal fuzzy numbers formed by experts' estimates. To our mind, the representative of this collection, i.e., a trapezoidal fuzzy number such that the sum of distances between it and all other members of the given finite collection is minimal, is of particular interest.

A representative can be regarded as a kind of group consensus, but in that case the degrees of experts' importance are neglected. A representative is something like a standard for the members of the considered collection. As the weights of physical bodies are measured by comparing them with the Paris standard kilogram, it seems natural for us to determine experts' weights of importance depending on how close experts' estimates are to a representative.

**4. Realization of proposed approach**

*DOI: http://dx.doi.org/10.5772/intechopen.93285*

*A New Approach for Assessing Credit Risks under Uncertainty*

*<sup>R</sup>*~*ij* � � <sup>¼</sup>

*j*th parameter given by the expert number *i*.

*j* n o <sup>¼</sup> *<sup>R</sup>*<sup>~</sup> <sup>∗</sup>

Step 1: *Compute the representative R*~ <sup>∗</sup> *of R*~<sup>0</sup>

following sequence: *R*~ <sup>∗</sup>

**Algorithm 1.**

Eq. (16).

approach described in [11].

concept's characteristics list.

"degree of credit risk":

**189**

*regulation R*~<sup>0</sup>

trapezoidal fuzzy numbers.

*j*

*expert by ω<sup>j</sup> and the final result by R*~.

Step 2: *Do* Step 3 *for j* ¼ 1, *m.* Step 3: *Compute* <sup>Δ</sup> *<sup>j</sup>* <sup>¼</sup> *<sup>ρ</sup> <sup>R</sup>*<sup>~</sup> <sup>∗</sup>

• If at least one <sup>Δ</sup><sup>j</sup> = 0 then *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> *<sup>R</sup>*<sup>~</sup> <sup>∗</sup>

Let *m* experts evaluate the values of *n* parameters in the form of trapezoidal fuzzy numbers. As a result, we get a rectangular matrix of dimension *m* � *n*:

The *i-*th column of the obtained matrix represents a collection of estimates of the

To assess the values of the parameters according to the credit risk criterion, we

n o, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … . *Denote the aggregation weight of the j-th*

*j*

find a representative of each column of the matrix. As a result, we obtain the

<sup>2</sup> , … , *<sup>R</sup>*<sup>~</sup> <sup>∗</sup> *n :* Now we present an algorithm for finding a representative of finite collection of

Step 0: *Initialization: the finite collection of trapezoidal fuzzy numbers R*~ *<sup>j</sup>*

;

• If Δ<sup>j</sup> > 0 for all j then compute ω<sup>j</sup> by Eq. (15) and obtain the final result by

In almost any field, you can get a rating scale using the following principles:

a. Define a list of characteristics by which the concept (object) is evaluated.

characteristic (in the original, specific numerical closed intervals were used).

The collection of ratings on the scale was called the *profile* of the concept. Since

b. Find polar characteristics in this list and form a polar scale.

c. At the poles, determine to what extent the concept possesses this

the gradation values of the scale are approximate (expert opinions), with the exception of the assigned pole values, the profile represents a fuzzy set of the

In the introduction, we mentioned the concept of a linguistic variable. This concept plays an important role in our study. Let us introduce the linguistic variable

For what follows, we need to determine a scale that can "measure" the opinions of experts regarding the risk of bankruptcy of the borrower. We use the general

<sup>1</sup> , *<sup>R</sup>*<sup>~</sup> <sup>∗</sup>

, *R*~ *<sup>j</sup>* � �: , *i* ¼ 1, *n*, *j* ¼ 1, *n:* (17)

n o, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>m</sup>*, *<sup>m</sup>* <sup>¼</sup> 2, 3, … *by Eq.* (14)*.*

� �*, its*

*R*~<sup>11</sup> *R*~<sup>12</sup> ⋯ *R*~1*<sup>n</sup> R*~<sup>21</sup> *R*~<sup>22</sup> ⋯ *R*~2*<sup>n</sup>* ⋯ ⋯⋯⋯ *R*~*m*<sup>1</sup> *R*~*m*<sup>2</sup> ⋯ *R*~*mn*

Thus, the main idea of the proposed method reduces to the following. The weight of importance for each expert is determined by a function inversely proportional to the distance between his estimate and the representative of the finite collection of all experts' estimates, i.e., the smaller the distance between an expert's estimate and the representative, the larger the weight of his importance.

Let *<sup>R</sup>*<sup>~</sup> *<sup>j</sup>*, *<sup>j</sup>*∈f g 1, 2, … , *<sup>m</sup>* , *<sup>m</sup>* <sup>¼</sup> 2, 3, … be a trapezoidal fuzzy number representing the *j*th expert's subjective estimate of the rating to an alternative under a given criterion. Estimates of all experts form the finite collection of trapezoidal fuzzy numbers *R*~ *<sup>j</sup>* � �. By Definition 4 and formula (14), we find the regulation *R*~<sup>0</sup> *j* n o and the representative *R*~ <sup>∗</sup> of this collection. Denote the *j*th expert's aggregation weight (weight of importance) and the final result of aggregation by *<sup>ω</sup><sup>j</sup>* and *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> *<sup>a</sup>*~, <sup>~</sup> *b*,~*c*, ~ *d* � �, respectively.

By the above reasoning, the weights and the final result of aggregation can be defined as follows:

$$\rho\_{j} = \frac{\left(\rho\left(\bar{\boldsymbol{\mathcal{R}}}^{\*}, \boldsymbol{\tilde{\mathcal{R}}}\_{j}\right)\right)^{-1}}{\sum\_{j=1}^{m} \left(\rho\left(\bar{\boldsymbol{\mathcal{R}}}^{\*}, \boldsymbol{\tilde{\mathcal{R}}}\_{j}\right)\right)^{-1}}, \ m = 2, 3, \ldots \tag{15}$$

and

$$\tilde{R} = \sum\_{j=1}^{m} \left( \alpha\_j \otimes \tilde{R}\_j \right) \tag{16}$$

It is obvious that P*<sup>m</sup> <sup>j</sup>*¼<sup>1</sup>*ω<sup>j</sup>* <sup>¼</sup> 1. In [7] it is proved that the function in expression (15) is always continuous (the denominator does not turn into 0 in any case).

In the following proposition and its corollaries, the properties and values of the aggregation result for special cases are established.

**Proposition 1** [7]**.** *For any finite collection of trapezoidal fuzzy numbers R*~ *<sup>j</sup>* � �, *j* ¼ 1, *m*, *m* ¼ 2, 3, … , *the following holds:*


**Corollary 1.** *If for all t, j* <sup>∈</sup> {1,2, … ,*m*} *<sup>R</sup>*~*<sup>t</sup>* <sup>¼</sup> *<sup>R</sup>*<sup>~</sup> *<sup>j</sup>* ) *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> *<sup>R</sup>*<sup>~</sup> <sup>∗</sup> . **Corollary 2.** *If the all estimates are identical then ω<sup>j</sup>* = 1/*m*.

**Summary**. The section introduces the approach to assessment of the credit risk. It also considers the rationale for the choice of fuzzy trapezoidal numbers as a form for the presentation of experts' estimates. The section contains important formalisms for determining the degrees of experts' importance and the result of aggregation of experts' estimates.
